Winter 2016 Schedule

January 13

Speaker: Nathan Schley

Title: Introduction to Quadratic Forms from an Algebraic Point of View

Abstract: I noticed when Justin gave his talk on abstract Witt rings that a foundational talk about quadratic forms and the Witt ring might be beneficial to many of you. In an attempt to give back to all of you for providing me with so many talks, as well as to have an opportunity to review the material myself, I will be covering the basics of quadratic forms as presented in Lam's Algebraic Theory of Quadratic forms. This will start with the polynomial definition everyone is aware of, quickly move this to a coordinate free definition accompanied by some special properties and the unique decomposition theorems. This structure along with two basic operations on these objects motivates the construction of the Witt ring, which will also be covered. I plan to end by covering some applications of these results and some proofs that I think demonstrate the flavor of the subject from what I have seen so far.

January 20

Speaker: Nadir Hajouji

Title: Primary Decomposition and Associated Primes

Abstract: Primary ideals and submodules (which will be defined in the talk) are a natural generalization of prime ideals that one encounters in algebraic geometry and commutative algebra. In nice situations (Noetherian - everything is Noetherian) one can "factor" ideals and submodules into primary components. This gives rise to the theory of associated primes, which eventually get generalized to associated points of schemes in algebraic geometry.

Since associated points keep coming up in the algebraic geometry course (that many of us are taking), but Vakil refuses to talk about primary ideals/submodules, I decided to give a talk outlining the classical theory of primary decomposition of ideals/submodules.

January 27

Speaker: Cindy Tsang

Title: Classical Iwasawa Theory and Z_p-Extensions

Abstract: The class number of a number field is an object of interest in Number Theory. In the late 1950's, Iwasawa initiated the study of the growth of the class numbers in certain towers of number fields, namely K_0\subset K_1\subset K_2\subset\cdots such that their union is Galois over K_0 with Galois group isomorphic to Z_p. We will discuss the proof of Iwasawa's theorem and some related problems concerning Z_p-extensions.

February 3

Speaker: David Wen

Title: All the Singularities: Put a Ring on it

Abstract: Whilst learning the ways of Algebraic Geometry, two main "benefits" eventually reveal themselves. The first. which is immediate, is working with a geometric space that is interwoven with it's functions field. Where "doing" algebra on the function field translates over to "doing" geometry on the space. The second not so immediate but eventual realization is that the "algebrafication" of the geometric spaces allows for the existence of singularities. Furthermore it allows for the management, understanding and partial classification of the singularities that appear. This talk will be a quick, and maybe dirty, introduction into singularities in Algebraic Geometry, with an emphasis on the types of singularities that appear on surfaces in the Minimal Model Program.

February 10

Speaker: Cindy Tsang

Title: Introduction to Abelian Class Field Theory

Abstract: The goal of class field theory is to characterize the finite Galois extensions of a field in terms of the arithmetic of the field itself. For abelian extensions over a number field, such a characterization is known. The goal of this talk is to explain what the characterization is. Specializing to Q, we will recover the Kronecker-Weber theorem, which states that every finite abelian extension of Q is contained in a cyclotomic extension.

February 17

No Seminar

February 24

No Seminar

March 2

No Seminar

March 9

Speakers: Nadir Hajouji and Steve Trettel

Title: Algebraic? Geometry?

Abstract: This talk came out of an attempt earlier this quarter to try and understand the principal congruence subgroups of SL(2,Z). Inspired by the ideas of covering space theory, we can associate a hyperbolic surface to each principal congruence subgroup, in the hopes of translating algebraic questions to geometric ones. However, during this translation process (initiated by Steve) algebraic questions that were interesting in their own right arose (solved by Nadir) and so we figured we would tell you all the cool stuff we learned working together!